Properties

Label 2-192-1.1-c1-0-2
Degree $2$
Conductor $192$
Sign $1$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s − 4·11-s + 2·13-s + 2·15-s + 2·17-s + 4·19-s − 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 4·33-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s + 2·45-s − 7·49-s + 2·51-s + 2·53-s − 8·55-s + 4·57-s − 4·59-s + 2·61-s + 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s + 0.280·51-s + 0.274·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $1$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.524886838\)
\(L(\frac12)\) \(\approx\) \(1.524886838\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.75443438739266983517275434456, −11.58676923620628310825913776398, −10.20677110069933956172337352464, −9.785656483951954927896383611288, −8.456614023795656741730916500766, −7.61947548271568794045481490015, −6.16826675668887144123811676692, −5.14444399128618341481533912587, −3.43299575130251299450107440988, −1.99999348241168201324552330088, 1.99999348241168201324552330088, 3.43299575130251299450107440988, 5.14444399128618341481533912587, 6.16826675668887144123811676692, 7.61947548271568794045481490015, 8.456614023795656741730916500766, 9.785656483951954927896383611288, 10.20677110069933956172337352464, 11.58676923620628310825913776398, 12.75443438739266983517275434456

Graph of the $Z$-function along the critical line